A lemma of Krull Intersection Theorem.

Question

This lemma is from Hungerford's Algebra (p.388, Lemma VIII.4.3).

Lemma VIII.4.3 Let $P$ be a prime ideal in a commutative ring $R$ with identity. If $C$ is a $P$-primary submodule of the Noetherian $R$-module $A$, then there exists a positive integer $m$ such that $P^m A\subseteq C$.

Proof: Let $I$ be the annihilator of $A$ in $R$ and consider the ring $\overline{R}=R/I$. Denote the coset $r+I\in \overline{R}$ by $\overline{r}$. Clearly $I\subseteq \{r\in R\mid rA\subseteq C\}\subseteq P$, whence $\overline{P}=P/I$ is an ideal of $\overline{R}$. $A$ and $C$ are each $\overline{R}$-modules with $\overline{r}a=ra$ ($r\in R, a\in A$). We claim that $C$ is a primary $\overline{R}$-submodule of $A$.

If $\overline{r}a\in C$ with $r\in R$ and $a\in A-C$, then $ra\in C$. Since $C$ is a primary $R$-submodule, $r^n A\subseteq C$ for some $n$, whence $\overline{r}^n A\subseteq C$ and $C$ is $\overline{R}$-primary. Since $\{\overline{r}\in \overline{R}\mid \overline{r}^k A\subseteq C\text{ for some }k>0\}=\{\overline{r}\in \overline{R}\mid r^k A\subseteq C\}=\{\overline{r}\in \overline{R}\mid r\in P\}=\overline{P}$, $\overline{P}$ is a prime ideal of $\overline{R}$ and $C$ is a $\overline{P}$-primary $\overline{R}$-submodule of $A$. (see Theorems VIII.2.9 and VIII.3.2).

Since $\overline{R}$ is Noetherian by Lemma VIII.4.2, $\overline{P}$ is finitely generated by Theorem VIII.1.9. Let $\overline{p}_1, ..., \overline{p}_s$ ($p_i\in P$) be the generators of $\overline{P}$. For each $i$ there exists $n_i$ such that $\overline{p}_i^{n_i}A\subseteq C$. If $m=n_1+\cdots+n_s$, then it follows from Theorems III.1.2(v) and III.2.5(vi) that $\overline{P}^m A\subseteq C$. The facts that $\overline{P}=P/I$ and $IA=0$ now imply that $P^m A\subseteq C$.

My Questions

1. I guess the assertion "We claim that $C$ is a primary $\overline{R}$-submodule of $A$" was used to prove that "For each $i$ there exists $n_i$ such that $\overline{p}_i^{n_i}A\subseteq C$". But I think $\overline{p}_i^{n_i}A\subseteq C$ can be obtained from $p_i\in P$ directly.
   My explaination: Since $p_i\in P$ and $C$ is a $P$-primary submodule of $A$, we have $p_i^{n_i} A\subseteq C$ for some $n_i$. It follows that $\overline{p}_i^{n_i} A=p_i^{n_i} A\subseteq C$.
   That is, I don't understand why the author spent a bunch of times to prove the unnecessary assertion.

2. Where is the condition "$IA=0$" used?

3. There is a typo in the sentence "whence $\overline{r}^n A\subseteq C$ and $C$ is $\overline{R}$-primary". It should be "$\overline{P}$-primary"

Answer 1

This lemma was used to proved the Krull Intersection Theorem.

The proof of Krull Intersection Theorem.

• Milne's A Primer of Commutative Algebra. Theorem 3.16 on page 13.
• Hungerford's Algebra. Theorem 4.4 on page 389.

The proof of Krull Intersection Theorem by using the Artin-Rees Lemma.

• Atiyah's Introduction to Commutative Algebra. Theorem 10.17 on page 110.
• Eisenbud's Commutative Algebra. Corollary 5.4 on page 150.
• Bosch's Algebraic Geometry and Commutative Algebra. Theorem 2 on page 71.
• Matsumura's Commutative Ring Theory. Theorem 8.9 on page 60.
• The Stack Project. Lemma 10.50.4. https://stacks.math.columbia.edu/tag/00IP

The proof of Krull Intersection Theorem by using the Primary Decomposition Theorem.

• Sharp's Steps in Commutative Algebra. Corollary 8.25 on page 159.
• Milne's A Primer of Commutative Algebra. Remark 19.14 on page 92.